CCSSM HS Algebra – Seeing Structure in Expressions
A-SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is
not 1), and use the formula to solve problems.
Example from Algebra Form and Function (McCallum, Connally, Hughes-Hallett et al)
p. 453 - 454
A person saving for retirement deposits $5000 every year into an IRA that pays 6% interest
per year, compounded annually. After the first deposit (but before any interest has been
earned) the balance in the account in dollars is
B1 = 5000
After 1 year has passed, the first deposit has earned interest, so the balance become
5000(1.06). Then the second deposit is made and the balance becomes
B2 = 5000 + 5000(1.06)
At the end of the second year, the balance earns interest so the interest total is [5000 +
5000(1.06)](1.06) which can be written as 5000(1.06) + 5000(1.06)2
The third deposit is made, so
B3 = 5000 + 5000(1.06) + 5000 (1.06)2
How much money is in the IRA at the start of year 6, right after a deposit is made?
How much money is in the IRA at the start of year 26, right after a deposit is made?
Solution:
For the answer to the value of the IRA after year 6, students can churn out the amount on a
calculator by noticing the structure of the equation
B6 = 5000 + 5000(1.06) + 5000(1.06)2 + 5000(1.06)3 + 5000(1.06)4 + 5000(1.06)5
B6 = $34,876.59
However, the calculations become very tedious for B26. Fortunately there is an algebraic
shortcut. We use the trick of multiplying both sides of the equation by the rate (1.06) and
adding the original value. Let’s do that for B6 to see how it works:
B6 = 5000 + 5000(1.06) + 5000(1.06)2 + 5000(1.06)3 + 5000(1.06)4 + 5000(1.06)5
1.06B6 + 5000 = 1.06[5000 + 5000(1.06) + 5000(1.06)2 + 5000(1.06)3 + 5000(1.06)4 + 5000(1.06)5] + 5000
Notice that the right hand side of the equation and the formula for B6 have almost every term
in common. If we distribute the 1.06 to each term in the brackets, we have
1.06B6 + 5000 = 5000(1.06) + 5000(1.06)2 + 5000(1.06)3 + 5000(1.06)4 + 5000(1.06)5 + 5000 (1.06)6 + 5000
Move the 5000 at the end of the right side to the beginning of the right side and we have
1.06B6 + 5000 = 5000 + 5000(1.06) + 5000(1.06)2 + 5000(1.06)3 + 5000(1.06)4 + 5000(1.06)5 + 5000 (1.06)6
1.06B6 + 5000 = B6 + 5000(1.06)6
Move the B6 terms to the left side and move the constant terms the right side
1.06B6 - B6 = 5000(1.06)6 – 5000
Solve for B6
. 06𝐵6 = 5000 (1.06)6 − 5000
𝐵6 =
5000 (1.06)6 − 5000
. 06
𝐵6 = $34876.59
Use the same logic to figure out the formula for the value after the 26th payment.
The summation form for a geometric series is
𝑖
𝑆𝑛 = 𝑎 + 𝑎𝑟 + 𝑎𝑟 2 + ⋯ + 𝑎𝑟 𝑛−1 = ∑𝑛−1
𝑖=0 𝑎𝑟
Notice that Sn has n terms, since a = ar0, so if we want to use the summation capability in a
calculator, to find S26, the i goes from 0 to 25. In the case of the above problem, a = 5000 and r
= 1.06